Several complete textbooks of mathematics on geometric optimal control theory exist in the literature, but little has been done with relevant applications in control engineering. This monograph is intended to fill this gap. It is based on graduate courses for mathematicians and physicists and presents results from two research projects in space mechanics and quantum control. The main topics developed in this book are: * Geometric optimal control theory: Pontryagin Maximum Principle and second order necessary and sufficient optimality conditions * Extensions of Riemannian geometry in optimal control theory * Optimal control in space mechanics * Application to the orbital transfer between elliptic orbits in the two and three body problem * Optimal control of dissipative quantum systems * Application in Nuclear Magnetic Resonance and Magnetic Resonance Imaging. The presentation is self-contained and readers can use our techniques to perform similar analysis in their own problems. Numerical tools have been developed in parallel during the research projects (shooting and continuation methods) and the codes are freely available at http://apo.enseeiht.fr/hampath/contacts.html